Shell average and harmonic function

Question

Let $d>0$ be fixed. For $r>0$, let $A(x,r)$ denote the annulus or solid shell centered at $x \in R^n$ with inner radius $r$ and outer radius $r+d$. The volume of $A(x,r)$ is denoted by $|A(x,r)|$. Let $f$ be a continuous function on $R^n$ such that $$\frac 1{|A(x,r)|}\int_{A(x,r)} f(y)\, dy = f(x),$$ for any $r>0$ and any $x \in R^n$. Is it true that $f$ is a harmonic function? If it is harmonic, how to show it ? Any reference will also be helpful.